Step 1: Define the Linear Function

Given:

• Initial population $P_0 = 75,000$
• Growth rate $r = 3,000$ people per year
• Time $t$ is the number of years since the model began

The population $P(t)$ as a function of time $t$ can be expressed as a linear equation:

$P(t) = r \cdot t + P_0$

Substituting the given values:

$P(t) = 3,000 \cdot t + 75,000$

Step 2: Find the $t$-Intercept

The $t$-intercept occurs when the population $P(t) = 0$. To find this:

$0 = 3,000 \cdot t + 75,000$

Solve for $t$:

$3,000 \cdot t = -75,000$ $t = \frac{-75,000}{3,000} = -25$

So, the $t$-intercept is $t = -25$. In coordinate form:

$(t, P) = (-25, 0)$

Interpretation of the $t$-Intercept

The $t$-intercept represents the time when the population would have been zero. So, 25 years before the tracking of the population began, the population would have been zero.

Step 3: Find the $P$-Intercept

The $P$-intercept occurs when $t = 0$, which is the initial population:

$P(0) = 3,000 \cdot 0 + 75,000 = 75,000$

So, the $P$-intercept is $P = 75,000$. In coordinate form:

$(t, P) = (0, 75,000)$

Interpretation of the $P$-Intercept

The $P$-intercept represents the population at the beginning of tracking, which was 75,000.

Summary

• The linear function is $P(t) = 3,000 \cdot t + 75,000$.
• The $t$-intercept is $(t, P) = (-25, 0)$, meaning the population would have been zero 25 years before tracking began.
• The $P$-intercept is $(t, P) = (0, 75,000)$, representing the population at the beginning of tracking, which was 75,000.

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Would you like more details on any part of the solution? Here are some related questions:

1. How would the function change if the growth rate were different?
2. What is the population after 10 years?
3. How would you model this if the growth rate were not constant?
4. How does the slope of the graph represent the rate of population growth?
5. What would the population be in 50 years?

Tip: In linear functions, the slope tells you the rate of change (e.g., growth or decline), and the intercept tells you the starting point.